# Tight Bounds for Asynchronous Collaborative Grid Exploration

## TL;DR

This paper establishes tight bounds on the minimum number of asynchronous mobile agents needed to locate a target cell in a grid, considering the presence or absence of a global compass, advancing understanding of distributed search algorithms.

## Contribution

It provides the first tight bounds for asynchronous grid exploration with finite automaton agents, highlighting the impact of having a global compass.

## Key findings

- Tight lower bounds for agents with a global compass.
- Tight bounds for agents without a global compass.
- Asynchronous environment increases exploration difficulty.

## Abstract

Consider a small group of mobile agents whose goal is to locate a certain cell in a two-dimensional infinite grid. The agents operate in an asynchronous environment, where in each discrete time step, an arbitrary subset of the agents execute one atomic look-compute-move cycle. The protocol controlling each agent is determined by a (possibly distinct) finite automaton. The only means of communication is to sense the states of the agents sharing the same grid cell. Whenever an agent moves, the destination cell of the movement is chosen by the agent's automaton from the set of neighboring grid cells. We study the minimum number of agents required to locate the target cell within finite time and our main result states a tight lower bound for agents endowed with a global compass. Furthermore, we show that the lack of such a compass makes the problem strictly more difficult and present tight upper and lower bounds for this case.